I make my brother (berkeleysprings) do the deed. I tried shooting his shotgun once just to know that I could and I almost dislocated my shoulder and was deaf for ten minutes, so I don't know if that would be my choice. I shot a raccoon once with a handgun (a rampaging raccoon that killed many of my chickens) and would possibly be able to do that with a chicken if I had too.
If you are attached to your chickens, it's a very difficult thing so if you can get someone else to do it for you I think that is your best bet. If not, you just have to realize that as difficult as it will be for you to do, it is the right thing to do in the end. I cried after I shot the raccoon and felt really bad about it, but it was killing my chickens (who are now locked up in fort knox) and it was something that I had to do. So if you have to cull your chicken yourself, realize that no method will be easy, you will probably cry and feel awful about it, but in the end it is the best thing for the animal.